There are $72$ boys and $90$ girls on the math team. For the next math competition, Mr. Johnson would like to arrange all of the students in equal rows with only girls or only boys in each row. What is the greatest number of students that can be in each row?
Solution: In order to know how many students Mr. Johnson can have in each row, we need a number that is a factor of ${72}$ and ${90}$, so that the ${72}$ boys and the ${90}$ girls can be divided up into equal rows. So, if each row had $\gray{9}$ students, there would be ${90} \div \gray{9} =10$ rows of girls and ${72} \div \gray{9} = 8$ rows of boys. This creates equal rows, but it isn't the greatest number of students per row! To find the greatest number of students, we want to find the greatest common factor of ${90}$ and ${72}$. To do so, let's find factors of ${90}$ and ${72}$. ${90}$ : $1, 2, 3,5, 6, 9, 10, 15, {18}, 30, 45, 90$ ${72}$ : $1, 2, 3,4,6, 8, 9, 12, {18}, 24, 36, 72$ The greatest common factor of ${90}$ and ${72}$ is ${18}$. In math notation this looks like: $ \text{gcf}({90}, {72}) = {18}$. The greatest number of students that can be in each row is ${18}$.